Patrick Gérard (Université Paris-Scalay

An explicit formula for the Benjamin-Ono equation.

Abstract:

The Benjamin-Ono equation is an evolution PDE in one space dimension introduced by T.B. Benjamin in 1967 in order to model long internal gravity waves in a two-layer fluid with infinite depth. As an important feature, it admits a Lax pair involving Toeplitz operators on the Hardy space. I will explain how this structure leads to an explicit formula for a general solution in terms of its initial datum, in both cases of periodic boundary conditions and of solutions on the line decaying at infinity. Applications to low regularity wellposedness and to the zero dispersion limit will be discussed.